However, the main source of inspiration for planar graphs was the Four Color Conjecture (now Theorem; cf. Chapter 8) that the vertices of any planar graph can be colored with four colors in such a way ...

A planar graph can have many combinatorial embeddings and each combinatorial embedding can yield many planar drawings of the graph. Each combinatorial embedding partitions the plane into a set of ...

Now imagine that you want to insert a new edge connecting two nodes in a planar graph, say nodes 1 and 6 in the example below. To do so, you’re going to perform a series of flips. From the starting ...

The graph below shows the total number of publications each year in Planar Graph Algorithms and Distance Queries. References [1] Planar graphs, negative weight edges, shortest paths, and near ...

A classic example, the Four Colour Theorem, illustrates a fundamental property of planar graphs, while recent advancements have extended these ideas to more specialised variants, including list ...

Faces are areas that are enclosed by edges in a planar graph. This includes both within and outside the areas. The image above illustrates what is meant by enclosed by edges ... Worked Example . If a ...

Problem 1: Find a subquadratic algorithm for testing if a given graph is 1-planar. It is also easy to check in cubic time if G is 2-apex by testing planarity of all 2-vertex-deleted subgraphs. More ...

But the three utilities problem isn't a puzzle so much to be solved, as rather an example of how some kinds of graph networks are not planar – that is, capable of having edges (lines) connecting their ...

The study of such graphs is called graph theory. Engineers need to find planarity in a graph when, for example, they are designing a computer chip without a crossed wire.

Graph partitioning is an essential preprocessing step in distributed graph computation and scientific simulations. Existing well-studied graph partitioners are designed for static graphs, but ...